Dipole Field Research Articles

Accreting Neutron Stars in 3D General-relativistic Magnetohydrodynamic Simulations: Jets, Magnetic Polarity, and the Interchange Slingshot

Abstract Accreting neutron stars differ from black holes by the presence of the star’s own magnetic field, whose interaction with the accretion flow is a central component in understanding these systems’ disk structure, outflows, jets, and spin evolution. It also introduces an additional degree of freedom, as the stellar dipole can have any orientation relative to the inner disk’s magnetic field. We present a suite of 3D general-relativistic magnetohydrodynamic simulations in which we investigate the two extreme polarities, with the dipole field being either parallel or antiparallel to the initial disk field, in both the accreting and propeller states. When the magnetosphere truncates the disk near or beyond the corotation radius, most of the system’s properties, including the relativistic jet power, are independent of the star–disk relative polarity. However, when the disk extends well inside the corotation radius, in the parallel orientation the jet power is suppressed and the inner disk is less dense and more strongly magnetized. We suggest a physical mechanism that may account for this behavior—the interchange slingshot—and discuss its astrophysical implications. When the star is in the rapidly accreting regime, which in most cases will be associated with strong spin-up, we expect large observational differences between the two magnetic orientations. This may be reflected in increased variability as the accretion flow drags in successive magnetic structures of varying polarity.

High-resolution motion compensation for brain PET imaging using real-time electromagnetic motion tracking.

Substantial improvements in spatial resolution in brain positron emission tomography (PET) scanners have greatly reduced partial volume effect, making head movement the main source of image blur. To achieve high-resolution PET neuroimaging, precise real-time estimation of both head position and orientation is essential for accurate motion compensation. A high-resolution electromagnetic motion tracking (EMMT) system with an event-by-event motion correction is developed for PET-CTscanners. EMMT is comprised of a source, an array of sensors, and a readout electronic unit (REU). The source acts as a transmitter and emits an EM dipole field. It is placed in close proximity to the sensor array and detects changes in EM flux density due to sensor movement. The REU digitizes signals from each sensor and captures precise rotational and translational movements in real time. Tracked motion in the EMMT coordinate system is synchronized with the PET list-mode data and transformed into the scanner coordinate system by locating paired positions in both systems. The optimal rigid motion is estimated using singular value decomposition. The rigid motion and depth-of-interaction (DOI) parallax effect are corrected by event-by-event rebinning of mispositioned lines-of-response (LORs). We integrated the EMMT with our recently developed ultra-high resolution Prism-PET prototype brain scanner and a commercial Siemens Biograph mCT PET-CT scanner. We assessed the imaging performance of the Prism-PET/EMMT system using multi-frame motion of point sources and phantoms. The mCT/EMMT system was validated using a set of point sources attached to both a mannequin head and a human volunteer, for simulating multiframe and continuous motions, respectively. Additionally, a human subject for [18F]MK6240 PET imaging wasincluded. The tracking accuracy of the Prism-PET/EMMT system was quantified as a root-mean-square (RMS) error of 0.49 for 100 axial rotations, and an RMS error of 0.15 mm for 100 mm translations.The percent difference (%diff) in average full width at half maximum (FWHM) of point source between motion-corrected and static images, within a motion range of and 10 mm from the center of the scanner's field-of-view (FOV), was 3.9%. The measured recovery coefficients of the 2.5-mm diameter sphere in the activity-filled partial volume correction phantom were 23.9%, 70.8%, and 74.0% for the phantom with multi-frame motion, with motion and motion compensation, and without motion, respectively. In the mCT/EMMT system, the %diff in average FWHM of point sources between motion-corrected and static images, within a motion range of and 10 mm from the center of the FOV, was 14%. Applying motion correction to the [18F]MK6240 PET imaging reduced the motion-induced spill-in artifact in the lateral ventricle region, lowering its standardized uptake value ratio (SUVR) from 0.70 to 0.34. The proposed EMMT system is a cost-effective, high frame-rate, and none-line-of-sight alternative to infrared camera-based tracking systems and is capable of achieving high rotational and translational tracking accuracies for mitigating motion-induced blur in high-resolution brain dedicated PETscanners.

Impact of random geometric errors in an elliptical superconducting magnet with application to a compact heavy-ion synchrotron

A compact heavy-ion synchrotron is under development for next-generation cancer therapy. A superconducting magnet is designed with a main dipole field of 3.5 T and a field error less than 5 × 10−4 to maintain compactness. The coil is wound directly with monolithic Nb-Ti wire on a curved elliptical mandrel, comprising 22 laminated layers to achieve the desired magnetic field. Given the critical need for field quality, it is imperative to determine the tolerance of coil fabrication and to devise a method to eliminate field error during the design stage. This paper presents a numerical investigation of possible random geometric errors stemming from fabrication and assembly tolerances in a curved elliptical superconducting magnet. We first provide an analytical formulation derived in a complex plane to estimate the field error of a straight elliptical coil. We then conduct a Monte Carlo simulation to assess cases involving misalignment of individual wires and coil block sectors. Additionally, we compare simulation results with field measurements obtained from a short model magnet tested previously to predict potential random errors in manufacturing. Finally, we calculate the beam dynamic aperture to assess the effects of random geometric errors on beam loss using Monte Carlo simulation results. This method enables the prediction of tolerances for fabricating high-quality high-field magnets and aids in making design decisions concerning the utilization of active shim coils.

MeerKAT observations of pair-plasma induced birefringence in the double pulsar eclipses

ABSTRACT PSR J0737−3039A/B is unique among double neutron star systems. Its near-perfect edge-on orbit causes the fast spinning pulsar A to be eclipsed by the magnetic field of the slow spinning pulsar B. Using high-sensitivity MeerKAT radio observations combined with updated constraints on the system geometry, we studied the impact of these eclipses on the incident polarization properties of pulsar A. Averaging light curves together after correcting for the rotation of pulsar B revealed enormous amounts of circular polarization and rapid changes in the linear polarization position angle, which occur at phases where emission from pulsar A is partially transmitted through the magnetosphere of pulsar B. These behaviours confirm that the eclipse mechanism is the result of synchrotron absorption in a relativistic pair-plasma confined to the closed-field region of pulsar B’s truncated dipolar magnetic field. We demonstrate that changes in circular polarization handedness throughout the eclipses are directly tied to the average line of sight magnetic field direction of pulsar B, from which we unambiguously determine the complete magnetic and viewing geometry of the pulsar.

Numerical analysis of the fungibility of disclination dipole by dislocation array or dislocation monopole

Disclinations have been discovered and applied to a wide variety of materials, underscoring the importance of their analysis. Previous studies have demonstrated that a disclination dipole can be represented by a dislocation array in a deformation schematic and by a dislocation monopole in a stress state. Based on these observations, we propose that the elastic field of a disclination dipole can be substituted with that of a dislocation array or dislocation monopole, allowing the development of the disclination theory grounded in dislocation theory. In this study, we conducted a quantitative analysis of the elastic strain–energy of disclination dipoles, dislocation arrays, and dislocation monopoles using linear elasticity theory and molecular dynamics (MD) simulation. While linear elasticity theory facilitates a quantitative evaluation of the elastic field, it fails to account for out-of-plane deformation of the material. Therefore, MD analysis, which considers out-of-plane deformation, was employed to obtain versatile results applicable to a broader range of materials. Our findings from both theoretical analyses indicate that, in terms of strain–energy, a disclination dipole with a large separation can be represented by a dislocation array, whereas a dipole with a small separation can be represented by a dislocation monopole. This study newly performs strain–energy analysis of dislocations and disclinations using these two theoretical approaches.

Enhanced functionalities of isovalently substituted 0.67(Bi1-xLaxFe0.97Ga0.03O3)-0.33(BaTiO3) relaxor piezoceramics synthesized via air quenching

The isovalently substituted lead-free binary solid solution, 0.67(Bi1-xLaxFe0.97Ga0.03O3)-0.33(BaTiO3) (BF-BT) (x ≤ 0.07), has been synthesized by solid state ceramic method via air quenching sintering. FTIR and XRD along with Rietveld structure refinement confirmed a pure perovskite phase with a pseudocubic structure (space group Pm3‾m) for developed compositions. SEM revealed a dense microstructure with fine grain size (<1.5 μm) and minimal porosity. XPS analysis reveals predominantly Bi+3 and Fe+3 states, along with oxygen vacancies (VO··). Ferroelectric studies demonstrated a maximum remanent polarization (Pr) of 12.3 μC/cm2 and coercive field EC of 30.95 kV/cm for La3+ concentration of x = 0.03. Overall decrement in Pr can be attributed to dipolar defect-induced random fields reducing the activation barrier for domain nucleation. Leakage current measurement reveals improved resistivity (∼108 Ω cm) up to an applied field of 30 kV/cm. Dielectric measurements unveiled two frequency-dependent anomalies in temperature dependence above room temperature, indicative of diffuse phase transitions. The Vogel-Fulcher model depicted an increasing freezing temperature (from 465 K for x = 0.01–551 K for x = 0.07) and decreasing activation energy (from ∼0.61 eV to 0.07 eV) with increasing La3+ concentration. The estimated diffuseness parameter around ∼2 suggested relaxor behavior. Doping with La resulted in increased dielectric permittivity at 10 kHz (from 4100 to 24066), showcasing the efficacy of site engineering in augmenting the functional properties of BF-BT for high-temperature dielectric and transducer applications.

Determination of the rigidity of the geomagnetic cutoff and simulation of the motion of particles in the Earth’s magnetosphere

A method for determination of the geomagnetic cutoff rigidity is presented. The method is based on the tracing of charged particles in the Earth’s magnetic field using Buneman-Boris’ particle-in-cell method. The results of the verification of the method are presented: in particular, a comparison with theoretical calculations in an ideal dipolar field and with previous calculations made under the real field condition. The developed method has shown a high reliability proven by the replication of the known effects. In the dipolar approximation, it has shown high accuracy in comparison with the theoretical calculations. Typical pattern of geomagnetic cutoff penumbra is also reproduced.

A novel plasma source concept for negative ion generation in neutral beam injectors for fusion applications

Abstract A low-temperature plasma can very well be confined by a simple magnetic dipole, such as in the Van Allen belts of Earth’s magnetoshpere. This configuration can be reproduced in laboratory as a small experimental device, designed in such a way that the magnetic field lines remain within a vacuum-tight container and are virtually not intercepted by the container wall. In this paper we propose to use a dipole field for the realization of an efficient negative Ion source. To this purpose, we analyze the plasma confinement capabilities of such plasma source, in order to assess the equilibrium pressure and estimate the particle trajectories and drifts.&amp;#xD;

Energetics of Buchdahl stars and the magnetic Penrose process

Buchdahl star is the most compact object without an event horizon and is an excellent candidate for a black hole mimicker. Unlike black holes, rotating Buchdahl star can be over-extremal with respect to the black hole, sustaining a larger spin. We show that it can also develop an ergosphere above the threshold spin β>1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta > 1/\\sqrt{2}$$\\end{document}, which allows extraction of its rotational energy. Electromagnetic field around Buchdahl star is also expected to differ from that of black hole in both strength and topology. In this paper, we explore the energetics of Buchdahl star focusing on the magnetic Penrose process in the two magnetic field configurations, i.e., uniform and dipole. Below the threshold spin, Buchdahl star is expected to be quiet, while above the threshold it can be much more efficient than the black hole if a dipolar magnetic field is developed on its surface.

Determination of a small elliptical anomaly in electrical impedance tomography using minimal measurements

Abstract We consider the problem of determining a small elliptical conductivity anomaly in a unit disc from boundary measurements. The conductivity of the anomaly is assumed to be a small perturbation from the constant background. A measurement of voltage across two point-electrodes on the boundary through which a constant current is passed. We further assume the limiting case when the distance between two electrodes go to zero, creating a dipole field. We show that three such measurements suffice to locate the anomaly size and location inside the disc. Two further measurements are needed to obtain the aspect ratio and the orientation of the ellipse. The investigation includes the studies of the stability of the inverse problem and optimal experiment design.

Comprehensive investigations on spectral and temporal features of GX 5−1 using AstroSat observations

ABSTRACT Comprehensive spectrotemporal analyses of the Z-type neutron star low-mass X-ray binary GX 5−1 were performed using 10 broad-band observations from AstroSat/Soft X-ray Telescope and Large Area X-ray Proportional Counter (LAXPC) instruments. The LAXPC-20 hardness–intensity diagram showed horizontal and normal branches (HBs and NBs) of the Z track which exhibited secular motion. The time-averaged spectra in the energy range 0.7–25.0 keV could be fitted with the model combination – $\tt {constant}\, \times \, \tt {tbabs}\, \times \, \tt {edge}\, \times \, \tt {edge}\, \times \, \tt {thcomp}\, \times \, \tt {diskbb}$. This yielded $\Gamma \, \sim$ 2, $kT_{\mathrm{ e}}\, \sim$ 3.3 keV, and $F_{\mathrm{ disc}}$/$F_{\mathrm{ total}}\, \sim$ 0.8 indicating the soft/intermediate spectral state of the source during the observations. Flux-resolved spectral analysis revealed a positive correlation between $kT_{\mathrm{ in}}$ and $F_{\mathrm{ bol}}$. However, a negative correlation was observed between them in one of the NBs. Time-averaged temporal analysis revealed multiple HB oscillations (HBOs) and NB oscillations (NBOs), and peaked noise components in the $\sim$5–50 Hz range. Furthermore, flux-resolved temporal analysis showed that the frequency of the HBOs correlates positively whereas the strength of HBOs correlates negatively with $F_{\mathrm{ bol}}$, indicating their probable origin from the accretion disc. In contrast, the frequency and strength of NBOs remain fairly constant with $F_{\mathrm{ bol}}$, suggesting that they originate from a different region in the system. Using the relativistic precession model along with highest frequency of the HBO, the upper limits of the magnetic dipole moment ($\mu$) and field strength (B) at the poles of the neutron star in the system were found to be 25.60$\times \, 10^{25}$ G cm3 and 3.64$\times \, 10^{8}$ G, respectively, for $k_{\mathrm{ A}}$ = 1.

Vacancy-Induced Symmetry Breaking in Titanium Dioxide Boosts the Photocatalytic Hydrogen Production from Methanol Aqueous Solution.

The key to optimizing photocatalysts lies in the efficient separation and oriented migration of the photogenerated carriers. Herein, we report that breaking continuous TiO6 tetragonal (D4h) symmetry in titanium dioxide material by oxygen vacancy engineering could induce a dipole field within the bulk phase and thus facilitate the separation and transfer of photogenerated electron-hole pairs. After further loading of Cu single-atom co-catalysts, the obtained catalyst attained a hydrogen (H2) yield rate of 15.84 mmol g-1 h-1 and a remarkable apparent quantum yield of 12.67% at 385 nm from methanol aqueous solution. This catalyst also demonstrated impressive stability for at least 24 h during the photocatalytic tests. The innovative concept of producing dipole fields in semiconductors by breaking the crystal symmetry offers a new perspective for designing photocatalysts.

Effect of thin film thickness of ferrofluid on inclined bioconvective flow

This research examines the effect of variable thin film thickness on the bioconvective ferrofluid flow through a rotating sloped surface when a magnetic field (dipole) is present. Similarity transformation is used to standardize the current model's governing equations. By using a finite element method, the normalized nonlinear differential equations are numerically solved. Variable thickness of ferrofluid thin film and bioconvective parameters (bioconvective Lewis number, bioconvective Peclet number, and micro-organism concentration difference) yield the following results: velocity profiles (radial, tangential, and axial), gravity flow (drainage velocity and induced velocity), temperature profile, concentration profile and microorganism profile. In addition, this investigation examines the density of moving microorganisms as well as local heat and mass transport. The temperature, concentration, radial, axial, drainage, induced, and motile microbe velocity are all improved with increasing film thickness. Variable thickness increases local heat transfer whereas increasing Brownian motion and thermophoresis parameters decreases it. This study could be helpful for self-sterilizing applications and biomedical engineering.

Unveiling complex magnetic field configurations in red giant stars

The recent measurement of magnetic field strength inside the radiative interior of red giant stars has opened the way toward full 3D characterization of the geometry of stable large-scale magnetic fields. However, current measurements, which are limited to dipolar (ℓ = 1) mixed modes, do not properly constrain the topology of magnetic fields due to degeneracies on the observed magnetic field signature on such ℓ = 1 mode frequencies. Efforts focused toward unambiguous detections of magnetic field configurations are now key to better understand angular momentum transport in stars. We investigated the detectability of complex magnetic field topologies (such as the ones observed at the surface of stars with a radiative envelope with spectropolarimetry) inside the radiative interior of red giants. We focused on a field composed of a combination of a dipole and a quadrupole (quadrudipole) and on an offset field. We explored the potential of probing such magnetic field topologies from a combined measurement of magnetic signatures on ℓ = 1 and quadrupolar (ℓ = 2) mixed mode oscillation frequencies. We first derived the asymptotic theoretical formalism for computing the asymmetric signature in the frequency pattern for ℓ = 2 modes due to a quadrudipole magnetic field. To access asymmetry parameters for more complex magnetic field topologies, we numerically performed a grid search over the parameter space to map the degeneracy of the signatures of given topologies. We demonstrate the crucial role played by ℓ = 2 mixed modes in accessing internal magnetic fields with a quadrupolar component. The degeneracy of the quadrudipole compared to pure dipolar fields is lifted when considering magnetic asymmetries in both ℓ = 1 and ℓ = 2 mode frequencies. In addition to the analytical derivation for the quadrudipole, we present the prospect for complex magnetic field inversions using magnetic sensitivity kernels from standard perturbation analysis for forward modeling. Using this method, we explored the detectability of offset magnetic fields from ℓ = 1 and ℓ = 2 frequencies and demonstrate that offset fields may be mistaken for weak and centered magnetic fields, resulting in underestimating the magnetic field strength in stellar cores. We emphasize the need to characterize ℓ = 2 mixed-mode frequencies, (along with the currently characterized ℓ = 1 mixed modes), to unveil the higher-order components of the geometry of buried magnetic fields and to better constrain angular momentum transport inside stars.

Simulation of the response of a magnetic polymersome in an inhomogeneous magnetic field

Purpose. Identification of the behavior of a magnetic polymersome placed in an inhomogeneous field of a point dipole using coarse-grained molecular dynamics simulations.Methods. The system under the study is represented as a set of interacting particles of two types: polymeric particles simulating a double layer of an amphiphilic membrane, and magnetic nanoparticles located in the membrane layer. Polymeric particles interact through elastic potentials designed to maintain an equilibrium spherical vesicular geometry. Magnetic nanoparticles interact with each other and external fields as point dipoles. The excluded volume and steric interaction of magnetic particles with polymeric walls are modeled in the form of soft repulsion. The entire system is under isothermal conditions, and the model parameters are chosen according to typical interaction energies relative to thermal fluctuations. A model polymersome with a diameter of about 100 nm in an aqueous solution at 25°C is considered. An inhomogeneous magnetic field is created by a dipole of constant direction located at a fixed distance. The dynamics of the system is monitored by numerical solution of the equations of particles motion with introduced interactions and imposed conditions.Results. In numerical experiments, the response of the polymersome to a magnetic field was obtained for three values of the parameter describing the inhomogeneity of the field. The problem with a gradual increase in the strength (and its gradient) of the magnetic field near the polymersome is considered. Changes in the magnetization of the system are shown, and the redistribution of the concentration of nanoparticles is analyzed. A simulation of the situation when the center of the polymersome at the initial moment was placed at a point with a fixed value of the magnetic field for three cases of inhomogeneity of the field was carried out. A significant restructuring of the magnetic layer of the vesicle combined with movement of the entire capsule was detected. Estimates are made about the average velocity based on the displacement of the object.Conclusion. The model allows to evaluate the features of the combined magnetic, structural and mechanical response of the polymersome to an inhomogeneous field in the context of potential applications for controlled delivery of contents into cells.

Deciphering the Effect of Defect Dipoles on the Polarization and Electrostrain Behavior in Perovskite Ferroelectrics.

Defect dipoles are crucial for regulating electromechanical properties in piezoelectric ceramics, but their effects on polarization and electrostrain behaviors are still unclear. Here, a reasonable theoretical model is proposed and evidenced by experiments to address a long-standing puzzle of the relationship between the internal bias field and defect dipoles. By incorporating the additional polarization induced by defect dipoles, we refine the classical theory to account for the recently reported asymmetric giant-strain behaviors. Phase-field simulation reveals the electrostrain evolution in response to defect dipole elastic distortion and additional polarization. This work not only elucidates the effect of defect dipoles on polarization and electrostrain but also advances the theoretical understanding of defects in piezoelectrics.

Osmium(III) Acetylacetonate and Its Missing Polymorph: A Magnetic and Structural Investigation.

Despite the potential for their application, the magnetic behavior of complexes containing 4d and 5d metal ions is underexplored, evidencing the need for benchmark multi-technique studies on simple molecules. We report here a structural and magnetic study on osmium(III) acetylacetonate [Os(acac)3]. X-ray single crystal diffraction did not allow us to determine the structure of the β-polymorph of [Os(acac)3]. The combined magnetic (dc magnetic measurements on powder and cantilever torque magnetometry on single crystal) and spectroscopic (electron paramagnetic resonance, EPR) characterization is here used to provide further evidence that its structure is indeed the one of the orthorhombic "missing polymorph", analogous to the ruthenium(III) derivative. Our study shows that all acetylacetonate complexes of the eighth group of the periodic table show dimorphism and are isomorphic. The EPR characterization allowed the experimental assessment of the easy axis nature of the ground doublet and the determination of the first hyperfine coupling in an osmium complex. Torque magnetometry, applied here for the first time on an osmium-based molecule, determined the orientation of the easy axis along the pseudo C3 axis of the complex. Ac magnetometric measurements revealed in-field slow relaxation of the magnetization further slowed by the suppression of dipolar fields via magnetic dilution in the isostructural gallium(III) analogue.

Dipole field as charge-transfer bridge between Cu atomic clusters/PtCu alloy nanocubes and nitrogen-rich C3N5 for superior photocatalytic hydrogen evolution

Utilizing spontaneous polarization field to harness charge transfer kinetics is a promising strategy to boost photocatalytic performance. Herein, a novel Cu atom clusters/PtCu alloy nanocubes coloaded on nitrogen-rich triazole-based C3N5 (PtCu-C3N5) with dipole field was constructed through facile photo-deposition and impregnation method. The dipole field-drive spontaneous polarization in C3N5 acts as a charge-transfer bridge to promote directional electron migration from C3N5 to Cu atom clusters/PtCu alloy. Through the synergistic effects between Cu atom clusters, PtCu alloy and dipole field in C3N5, the optimized Pt2Cu3-C3N5 achieved a record-high performance with H2 formation rate of 4090.4 μmol g−1 h−1 under visible light, about 154.4-fold increase compared with pristine C3N5 (26.5 μmol g−1 h−1). Moreover, the apparent quantum efficiency was up to 25.33 % at 320 nm, which is greatly superior than most previous related-works. The directional charge transfer mechanism was analyzed in detail through various characterizations and DFT calculations. This work offers a novel pathway to construct high-efficiency multi-metal photocatalysts for solar energy conversion.

Columnar Macrocyclic Molecule Tailored Grain Cage to Stabilize Inorganic Perovskite Solar Cells with Suppressed Halide Segregation

AbstractSolidifying the soft lattice of all‐inorganic mixed‐halide perovskites is of great importance to restrain the notorious halide segregation under persistent light illumination. Herein, a multifunctional columnar macrocyclic molecule additive, namely cucurbituril into perovskite precursor to enhance the crystallization and reduce the defect density in the final perovskite film is introduced. Based on the theoretical calculation and simulation, the cucurbituril molecule with a strong double‐ended negatively‐charged cavity surrounded by terminated oxygen atoms not only coordinates with dangling Pb2+ ions to form host‐guest complexation but also induces an electric dipole field at perovskite grain boundary to effectively repel the iodide ion migration from the inside grain to the defective boundary, significantly suppressing the halide segregation and improving the device performance. As a result, the carbon‐based, all‐inorganic CsPbI2Br solar cell achieves an enhanced efficiency of 15.59% with great tolerance to environmental stresses. These findings provide new insights into the development of a novel passivation strategy with macrocyclic molecules for making high‐efficiency and stable perovskite solar cells.

A quantitative explanation of the cyclotron-line variation in accreting magnetic neutron stars of super-critical luminosity

Context. Magnetic neutron stars (NSs) often exhibit a cyclotron resonant scattering feature (CRSF) in their X-ray spectra. Cyclotron lines are believed to be generated in the radiative shock in the accretion column. High-luminosity NSs show a smooth anti-correlation between the cyclotron-line centroid (ECRSF) and X-ray luminosity (LX). Aims. It has been pointed out that the observed ECRSF − LX smooth anti-correlation in high-luminosity NSs is in tension with the theoretically predicted one, if the radiative shock is the site of cyclotron-line formation. The shock height increases approximately linearly with luminosity, while the dipolar magnetic field drops as the cubic power of distance, thereby implying that the cyclotron-line energy ought to decrease significantly when the luminosity increases by, say, an order of magnitude, which is contrary to observations. Since there is no other candidate site for the cyclotron-line formation, we re-examine the predicted rate of change of the cyclotron-line energy with luminosity at the radiative shock, taking a closer look at the Physics involved. Methods. We developed a purely analytical model describing the overall dependence of the observed cyclotron energy centroid on the shock front’s height, including both the relativistic boosting and the gravitational redshift effects in our considerations. The relativistic boosting effect is due to the mildly relativistic motion of the accreting plasma upstream with respect to the shock’s reference frame. Reults. We find that the cyclotron-line energy varies with (a) the shock height due to the dipolar magnetic field, (b) the Doppler boosting between the shock and bulk-motion frames, and (c) the gravitational redshift. We show that the relativistic effects noticeably weaken the predicted ECRSF − LX anti-correlation. We use our model to fit the data of the X-ray source V0332+53 that exhibits a weak negative correlation and demonstrate that the model fits the data impressively well, thereby alleviating the tension between observations and theory. Conclusions. The reported weak anti-correlation between cyclotron-line centroid and X-ray luminosity in the supercritical accretion regime may be explained by the combination of the variation of the magnetic-field strength along the accretion column, the effect of Doppler boosting, and the gravitational redshift. As a result of these effects, the actual magnetic field on the surface of the neutron star may be a factor of ∼2 larger than the naively inferred value from the observed CRSF.